A man runs along a horizontal road holding his umrella vertical in order to afford maximum protection form rain. The rain is actually.

Options

A. Falling vertical
B. Comming from front of the man
C. Coming from the back of the man
D. Either of (a), (b) or (c)

Explanation

To solve the problem, we need to analyze the situation using vector components. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
The man is running horizontally while holding his umbrella vertically. We need to determine the direction from which the rain appears to be falling relative to the man.

### Step 2: Define the Velocities
Let:
- \( v_m \) = velocity of the man (horizontal)
- \( v_r \) = velocity of the rain (vertical)

### Step 3: Set Up the Coordinate System
Assume the man is running to the right. We can define:
- The horizontal direction (to the right) as positive.
- The vertical direction (downward) as negative.

### Step 4: Analyze the Rain's Velocity Relative to the Man
As the man runs, he perceives the rain to be falling at an angle. The velocity of the rain relative to the man can be expressed as:
\[ v_{rm} = v_r - v_m \]
Here, \( v_{rm} \) is the velocity of the rain with respect to the man.

### Step 5: Determine the Resultant Velocity
To find the resultant velocity of the rain as observed by the man, we need to consider both the horizontal and vertical components:
- The vertical component remains \( v_r \) (downward).
- The horizontal component is \( -v_m \) (to the left, since the man is running to the right).

### Step 6: Calculate the Angle of the Resultant Velocity
The resultant velocity vector can be found using the Pythagorean theorem:
\[ v_{resultant} = \sqrt{v_r^2 + v_m^2} \]
The angle \( \theta \) at which the rain appears to fall can be calculated using:
\[ \tan(\theta) = \frac{v_r}{v_m} \]

### Step 7: Conclusion
Since the man is running horizontally, the rain appears to be falling at an angle towards him from the back. Therefore, the rain appears to be coming from behind him.

### Final Answer
The rain is actually falling vertically, but due to the man's horizontal motion, it appears to be coming from behind him.

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